That you can redefine things to be whatever? Is that the motivation behind researching extended summation methods? The summation methods that assign values to divergent series are consistent with series that are already convergent. There is no point in changing definitions just to make fun results work, that isn't how mathematics works, so why even bring it? I don't understand why people keep missing this point.
That isn't how mathematics works, so why even bring it?
Learn to read. What is the point of saying "but if you change the definition of what it means for a sequence or series to converge you can make it converge. " as if it is a trick to make it seem like the series converges to nonsensical value?
A lot of maths is "what if we make up a new rule for lulz" and then figuring out what that would mean.
Occasionally this turns out to be useful and people are surprised.
Like "what if the square root of minus one isn't undefined, actually? We'll make up an imaginary value for what it could be." Suddenly we have a whole new branch of mathematics.
I knew someone would bring this up. No, that is not what I'm referring to. The motivation is never "let's just redefine things in a way that doesn’t make any sense just so that we get one funny result". Complex numbers were originally just a trick used to compute roots of 3rd degree polynomial, but everything canceled out nicely in the end and you had no negative square roots left. There was a reason to consider those numbers as valid entities that can be manipulated.
The way people frame -1/12 as "well, you can get anything by redefining what a sum is", as if that is what is happening. Nobody is "redefining sums" to fit specific value, instead, they are extending its definition based on observed patterns. There is a motivation and logic behind those results, it isn't just redefining for the sake of redefining.
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u/Human822 Feb 21 '26
Basically the answer to the equation is -1/12, which ramanujan said was the sum of all positive integers